scholarly journals A note on N-soliton solutions for the viscid incompressible Navier–Stokes differential equation

AIP Advances ◽  
2022 ◽  
Vol 12 (1) ◽  
pp. 015308
Author(s):  
R. Meulens
Keyword(s):  
Differential Equation ◽  
Soliton Solutions ◽  
Navier Stokes

scholarly journals On the Use of Recursive Evaluation of Derivatives and Padé Approximation to Solve the Blasius Problem

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Asai Asaithambi
Keyword(s):  
Differential Equation ◽  
Computational Effort ◽  
Navier Stokes ◽  
Series Solutions ◽  
Third Order ◽  
Navier Stokes Equation ◽  
Order Problem ◽  
Algorithmic Differentiation ◽  
Blasius Problem ◽  
Recursive Evaluation

The Blasius problem is one of the well-known problems in fluid mechanics in the study of boundary layers. It is described by a third-order ordinary differential equation derived from the Navier-Stokes equation by a similarity transformation. Crocco and Wang independently transformed this third-order problem further into a second-order differential equation. Classical series solutions and their Padé approximants have been computed. These solutions however require extensive algebraic manipulations and significant computational effort. In this paper, we present a computational approach using algorithmic differentiation to obtain these series solutions. Our work produces results superior to those reported previously. Additionally, using increased precision in our calculations, we have been able to extend the usefulness of the method beyond limits where previous methods have failed.

Kink soliton solutions and bifurcation for a nonlinear space-fractional Kolmogorov–Petrovskii–Piskunov equation in circuitry, chemistry or biology

2019 ◽  
Vol 33 (30) ◽  
pp. 1950372
Author(s):  
Mei-Xia Chu ◽  
Bo Tian ◽  
Hui-Min Yin ◽  
Su-Su Chen ◽  
Ze Zhang
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Wave Velocity ◽  
Dynamic Systems ◽  
Traveling Wave ◽  
Soliton Solutions ◽  
Mutant Gene ◽  
Phase Portraits ◽  
Original Equation ◽  
Kink Soliton

Circuitry and chemistry are applied in such fields as communication engineering and automatic control, environmental protection and material/medicine sciences, respectively. Biology works as the basis of agriculture and medicine. Studied in this paper is a nonlinear space-fractional Kolmogorov–Petrovskii–Piskunov equation for the electronic circuitry, chemical kinetics, population dynamics, neurophysiology, population genetics, mutant gene propagation, nerve impulses transmission or molecular crossbridge property in living muscles. Kink soliton solutions are obtained via the fractional sub-equation method. Change of the fractional order does not affect the amplitudes of the kink solitons. Via the traveling transformation, the original equation is transformed into the ordinary differential equation, while we obtain two equivalent two-dimensional planar dynamic systems of that ordinary differential equation. According to the bifurcation and qualitative considerations of the planar dynamic systems, we display the corresponding phase portraits when the traveling-wave velocity is nonzero or zero. Nonlinear periodic waves of the original equation are obtained when the traveling-wave velocity is zero.

A probabilistic method for Navier-Stokes vortices

2001 ◽  
Vol 38 (04) ◽  
pp. 1059-1066
Author(s):  
Xinyu He
Keyword(s):  
Differential Equation ◽  
Stochastic Equation ◽  
Probabilistic Method ◽  
Complex Function ◽  
Linear Partial Differential Equation ◽  
Navier Stokes ◽  
Sample Paths ◽  
Vortex Centre ◽  
Suitable Notion ◽  
Moving Vortex

Consider a Navier-Stokes incompressible turbulent fluid in R 2. Let x(t) denote the position coordinate of a moving vortex with initial circulation Γ0 > 0 in the fluid, subject to a force F. Define x(t) as a stochastic process with continuous sample paths described by a stochastic differential equation. Assuming a suitable notion of weak rotationality, it is shown that the stochastic equation is equivalent to a linear partial differential equation for the complex function ψ, i∂ψ/∂t = [-Γ0Δ + F] ψ, where |ψ|2 = ρ(x,t), ρ being the probability density function of finding the vortex centre in position x at time t.

scholarly journals Complex Soliton Solutions to the Gilson–Pickering Model

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 18 ◽  
Author(s):  
Haci Mehmet Baskonus
Keyword(s):  
Differential Equation ◽  
Analytical Method ◽  
Soliton Solutions ◽  
2D And 3D

In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson–Pickering model is applied. A set of new complex soliton solutions to the Gilson–Pickering model are successfully constructed. In addition, 2D and 3D graphs and contour simulations to the complex soliton solutions are plotted with the help of computational programs. Finally, at the end of the manuscript a conclusion about new complex soliton solutions is given.

MULTI-SOLITON SOLUTIONS AND THEIR INTERACTIONS FOR THE (2+1)-DIMENSIONAL SAWADA-KOTERA MODEL WITH TRUNCATED PAINLEVÉ EXPANSION, HIROTA BILINEAR METHOD AND SYMBOLIC COMPUTATION

2009 ◽  
Vol 23 (25) ◽  
pp. 5003-5015 ◽  
Author(s):  
XING LÜ ◽  
TAO GENG ◽  
CHENG ZHANG ◽  
HONG-WU ZHU ◽  
XIANG-HUA MENG ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Symbolic Computation ◽  
Bilinear Form ◽  
Soliton Solutions ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Truncated Painlevé Expansion ◽  
Hirota Bilinear ◽  
Bilinear Equations

In this paper, the (2+1)-dimensional Sawada-Kotera equation is studied by the truncated Painlevé expansion and Hirota bilinear method. Firstly, based on the truncation of the Painlevé series we obtain two distinct transformations which can transform the (2+1)-dimensional Sawada-Kotera equation into two bilinear equations of different forms (which are shown to be equivalent). Then employing Hirota bilinear method, we derive the analytic one-, two- and three-soliton solutions for the bilinear equations via symbolic computation. A formula which denotes the N-soliton solution is given simultaneously. At last, the evolutions and interactions of the multi-soliton solutions are graphically discussed as well. It is worthy to be noted that the truncated Painlevé expansion provides a useful dependent variable transformation which transforms a partial differential equation into its bilinear form and by means of the bilinear form, further study of the original partial differential equation can be conducted.

Mechanics of a Free-Surface Liquid Film Flow

1987 ◽  
Vol 54 (4) ◽  
pp. 951-954 ◽  
Author(s):  
Cyrus K. Aidun
Keyword(s):  
Differential Equation ◽  
Free Surface ◽  
Film Flow ◽  
Navier Stokes ◽  
Experimental Measurements ◽  
Navier Stokes Equation ◽  
Higher Order Terms ◽  
Liquid Film Flow ◽  
Surface Liquid ◽  
Similar Equation

The mechanics of a free surface viscous liquid curtain flowing steadily between two vertical guide wires under the influence of gravity is investigated. The Navier-Stokes equation is integrated over the film thickness and an integro-differential equation is derived for the average film velocity. An approximate nonlinear differential equation, attributed to G. I. Taylor, is obtained by neglecting the higher order terms. An analytical solution is obtained for a similar equation which neglects the surface tension effects and the results are compared with the experimental measurements of Brown (1961).

scholarly journals An Axisymmetric Squeezing Fluid Flow between the Two Infinite Parallel Plates in a Porous Medium Channel

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
S. Islam ◽  
Hamid Khan ◽  
Inayat Ali Shah ◽  
Gul Zaman
Keyword(s):  
Differential Equation ◽  
Porous Medium ◽  
Fluid Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Parallel Plates ◽  
Transform Method ◽  
Navier Stokes Equations ◽  
Differential Transform ◽  
Fluid Parameters

The flow between two large parallel plates approaching each other symmetrically in a porous medium is studied. The Navier-Stokes equations have been transformed into an ordinary nonlinear differential equation using a transformationψ(r,z)=r2F(z). Solution to the problem is obtained by using differential transform method (DTM) by varying different Newtonian fluid parameters and permeability of the porous medium. Result for the stream function is presented. Validity of the solutions is confirmed by evaluating the residual in each case, and the proposed scheme gives excellent and reliable results. The influence of different parameters on the flow has been discussed and presented through graphs.

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